Methods for Two Categorical Variables – R x C Tables
In previous notes we could use either Fisher’s Exact Test or the Chi-square Test to analyze contingency
tables with two rows and two columns. However, to analyze contingency tables with r rows and c
columns (where r > 2 and c > 2) we can only use the Chi-square Test. Usually this is referred to as a Chisquare Test of Independence. This test is used to compare two or more population proportions.
Example: Prevention of deep venous thrombosis (DVT) is a critical issue in patients undergoing total hip
replacement surgery. Orthopedic surgeons recognize the importance of prophylaxis in the management
of their patients, but do not agree on an optimal method. In this study, three different prophylactic
measures are to be compared for the prevention of a proximal DVT after total hip replacement surgery.
Three independent groups of patients undergoing total hip replacements were given different
prophylactics. After surgery it was noted whether patients had complications from a proximal DVT or
not. The results are given in the contingency table below.
Group 1
Group 2
Group 3
Total
DVT Complication
76
71
69
216
No Complication
9
4
11
24
Total
85
75
80
240
Step 0: Define the research question
Are prophylactic measure and DVT status independent of one another? That is, is there a
relationship between prophylactic measure and DVT status?
Step 1: Determine the null and alternative hypothesis
H0: Prophylactic measure and DVT status are independent of one another (i.e. there is no
relationship)
Ha: Prophylactic measure and DVT status are not independent of one another (i.e. there is a
relationship)
Step 2: Calculate the test statistic and p-value
We will again use the Chi-square test statistic. Therefore, we must again find expected counts in order
to make sure the Chi-square test is valid. We’ll first have to find the overall percentage for DVT
complications and No complications.

DVT Complications 

No Complications 
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Once we have the overall percentages, we can find the expected counts for each cell in the contingency
table.
DVT Complication
Group 1
Total
85
Group 2
75
Group 3
80
Total
216
No Complication
24
240
We can use JMP to find the test statistic and p-value. We’ll have to enter the data into JMP as follows:
Then we’ll again choose Analyze  Fit Y by X, and put Prophylactic Measure in for X, Factor and DVT
Status in for Y, Response. You should then get the following output. You’ll again want to use the
Pearson test statistic and p-value.
Step 3: Report the conclusion in context of the research question.
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We can also learn about the relationship between the two variables by looking at the mosaic plot.
Example: A standardized procedure for determining a person’s susceptibility to hypnosis is the Stanford
Hypnotic Scale, Form C (SHSS:C). Recently, a new method called the Computer-Assisted Hypnosis Scale
(CAHS), which uses a computer as a facilitator of hypnosis, has been developed. Each scale classifies a
person’s hypnotic susceptibility as low, medium, or high. Researchers at the University of Tennessee
compared the two scales by administering both tests to each of 130 undergraduate volunteers
(Psychological Assessment, March 1995). The hypnotic classifications are summarized in the
contingency table given below.
SHSS:C Level
Low
Medium
High
Total
Low
32
11
6
49
CAHS Level
Medium
14
14
16
44
High
2
6
29
37
Total
48
31
51
130
Questions:
1. Looking at the mosaic plot given below. Does is appear there is a relationship between SHSS:C
level and CAHS level? Explain.
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2. Carry out the hypothesis test to determine whether there is a relationship between SHSS:C level
and CAHS level.
Step 0: Define the research question
Are SHSS:C level and CAHS level independent of one another? That is, is there a
relationship between SHSS:C level and CAHS level?
Step 1: Determine the null and alternative hypotheses
Step 2: Calculate the test statistic and p-value
Step 3: Report the conclusion in context of the research question
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Example: Boles and Johnson (Journal of Addictive Diseases 2001) examined the beliefs held by
adolescents regarding smoking and weight. Respondents characterized their weight into three
categories: underweight, overweight, or appropriate. Smoking status was characterized according to
the question “Do you currently smoke, meaning one or more cigarettes per day?” The data are given in
the table below.
Smoke
Do Not Smoke
Total
Underweight
17
97
114
Overweight
25
142
167
Appropriate
96
816
912
Total
138
1055
1193
Questions:
3. Create a mosaic plot of the data. Looking at the mosaic plot, does it appear there is a
relationship between an adolescent’s perception of weight and smoking status? Explain.
4. Carry out the hypothesis test to determine whether there is a relationship between perception
of weight and smoking status.
Step 0: Define the research question
Are perception of weight and smoking status independent of each other? That is, is
there a relationship between perception of weight and smoking status?
Step 1: Determine the null and alternative hypotheses
Step 2: Calculate the test statistic and p-value
Step 3: Report the conclusion in context of the research question
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Example: Gardemann et al. (1998) surveyed genotypes at an insertion/deletion polymorphism of the
apolipoprotein B signal peptide in 2,259 men. The data are given in the following table.
Ins/Ins
Ins/Del
Del/Del
Total
No Coronary
Artery Disease
268
199
42
509
Coronary Artery
Disease
807
759
184
1,750
Total
1,075
958
226
2,259
Questions:
5. Create a mosaic plot of the data. Looking at the mosaic plot, does it appear there is a
relationship between apolipoprotein B signal peptide and coronary artery disease? Explain.
6. Carry out the hypothesis test to determine whether there is a relationship between
apolipoprotein B signal peptide and coronary artery disease.
Step 0: Define the research question
Are apolipoprotein B signal peptide and coronary artery disease independent of each
other? That is, is there a relationship between apolipoprotein B signal peptide and
coronary artery disease?
Step 1: Determine the null and alternative hypotheses
Step 2: Calculate the test statistic and p-value
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Step 3: Report the conclusion in context of the research question
Example: A study was done to investigate the relationship between maternal drinking and congenital
malformation. After the first three months of pregnancy, the women in the sample completed a
questionnaire about alcohol consumption. Following childbirth, observations were recorded on
presence of congenital sex organ malformations. The data are given in the contingency table below.
Alcohol
Consumption
0
<1
1–2
3–5
≥6
Total
Malformation
Absent
Present
17,066
48
14,464
38
788
5
126
1
37
1
32,481
93
Total
17,114
14,502
793
127
38
32,574
Questions:
7. Create a mosaic plot of the data. Looking at the mosaic plot, does it appear there is a
relationship between day of the week and type of trap used? Explain.
8. Carry out the hypothesis test to determine whether there is a relationship between alcohol
consumption and malformation status.
Step 0: Define the research question
Are alcohol consumption and malformation status independent of each other? That is, is
there a relationship between alcohol consumption and malformation status?
Step 1: Determine the null and alternative hypotheses
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Step 2: Calculate the test statistic and p-value
Step 3: Report the conclusion in context of the research question
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